Angles in a Triangle Worksheets - Math Monks - Free Printable

Let's solve each problem step by step. The key concept here is that the sum of the interior angles in any triangle is always 180°. Additionally, if an exterior angle is shown, we can use the fact that:

- An exterior angle equals the sum of the two opposite interior angles.
- A straight line forms a 180° angle.

We'll go through each triangle one by one.

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1.

Given: 70°, 45°, find $ x^\circ $

$$
x = 180^\circ - 70^\circ - 45^\circ = 65^\circ
$$

✔ Answer: 65°

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2.

Given: 82°, 35°, find $ x^\circ $

$$
x = 180^\circ - 82^\circ - 35^\circ = 63^\circ
$$

✔ Answer: 63°

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3.

Given: 58°, 86°, find $ x^\circ $

$$
x = 180^\circ - 58^\circ - 86^\circ = 36^\circ
$$

✔ Answer: 36°

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4.

Given: 61°, 52°, find $ x^\circ $

Note: $ x^\circ $ is an exterior angle at the base.

First, find the interior angle adjacent to $ x^\circ $:
$$
\text{Interior angle} = 180^\circ - x^\circ
$$

But since the triangle has angles 61° and 52°, the third interior angle is:
$$
180^\circ - 61^\circ - 52^\circ = 67^\circ
$$

Now, $ x^\circ $ is the exterior angle at that vertex (adjacent to 67°), so:
$$
x = 180^\circ - 67^\circ = 113^\circ
$$

Alternatively, using exterior angle rule:
Exterior angle = sum of two non-adjacent interior angles
$$
x = 61^\circ + 52^\circ = 113^\circ
$$

✔ Answer: 113°

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5.

Given: 50°, $ y^\circ $, $ x^\circ $, and an exterior angle of 120°

The 120° is an exterior angle, so it equals the sum of the two opposite interior angles.

So:
$$
x + 50^\circ = 120^\circ \Rightarrow x = 70^\circ
$$

Now, in the triangle, total angles:
$$
x + 50^\circ + y = 180^\circ \Rightarrow 70^\circ + 50^\circ + y = 180^\circ \Rightarrow y = 60^\circ
$$

✔ Answer: $ x = 70^\circ $, $ y = 60^\circ $

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6.

Given: 77°, 64°, find $ x^\circ $

$$
x = 180^\circ - 77^\circ - 64^\circ = 39^\circ
$$

✔ Answer: 39°

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7.

Given: 13°, 29°, find $ x^\circ $

$$
x = 180^\circ - 13^\circ - 29^\circ = 138^\circ
$$

✔ Answer: 138°

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8.

Given: 64°, 38°, $ q^\circ $, and $ x^\circ $ is the exterior angle at the top.

First, find $ q^\circ $ — the interior angle at the bottom left:
$$
q = 180^\circ - 64^\circ - 38^\circ = 78^\circ
$$

Now, $ x^\circ $ is the exterior angle at the top vertex (opposite to $ q^\circ $). So:
$$
x = 180^\circ - q = 180^\circ - 78^\circ = 102^\circ
$$

Alternatively, use exterior angle rule:
Exterior angle = sum of two opposite interior angles
$$
x = 64^\circ + 38^\circ = 102^\circ
$$

✔ Answer: $ x = 102^\circ $, $ q = 78^\circ $

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9.

Given: 81°, 69°, find $ x^\circ $ and $ y^\circ $

First, $ x^\circ $ is an exterior angle at the bottom left.

So:
$$
x = 81^\circ + 69^\circ = 150^\circ
$$

Now, $ y^\circ $ is the interior angle at the top right.

Sum of triangle angles:
$$
y = 180^\circ - 81^\circ - 69^\circ = 30^\circ
$$

✔ Answer: $ x = 150^\circ $, $ y = 30^\circ $

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10.

Right triangle: 90°, 45°, find $ x^\circ $

$$
x = 180^\circ - 90^\circ - 45^\circ = 45^\circ
$$

✔ Answer: 45°

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11.

Given: 35°, 15°, find $ x^\circ $

This is a straight line at the base, but $ x^\circ $ is an exterior angle.

Wait — actually, look at the triangle: the 35° and 15° are two interior angles.

So:
$$
x = 180^\circ - 35^\circ - 15^\circ = 130^\circ
$$

But wait — the diagram shows $ x^\circ $ as an exterior angle at the bottom-left vertex.

So the interior angle at that vertex is $ x^\circ $'s supplement.

But let’s clarify:

If the triangle has angles 35°, 15°, then the third interior angle is:
$$
180^\circ - 35^\circ - 15^\circ = 130^\circ
$$

Then $ x^\circ $ is the exterior angle at that vertex, so:
$$
x = 180^\circ - 130^\circ = 50^\circ
$$

Wait — no! That would be if $ x $ were adjacent to the 130°.

But looking at the diagram: the triangle has angles 35°, 15°, and the third angle is unknown. But $ x^\circ $ is marked outside at the bottom-left, meaning it's the exterior angle at the vertex with the 35°?

Wait — recheck:

Actually, from the diagram:

- One angle is 35° (top)
- One angle is 15° (bottom-right)
- $ x^\circ $ is labeled at the bottom-left, and it's an exterior angle

So the interior angle at the bottom-left is adjacent to $ x^\circ $. Let’s call that interior angle $ a^\circ $.

Then:
$$
a + x = 180^\circ \quad \text{(straight line)}
$$

But in the triangle:
$$
a + 35^\circ + 15^\circ = 180^\circ \Rightarrow a = 130^\circ
$$

Then:
$$
x = 180^\circ - 130^\circ = 50^\circ
$$

✔ Answer: 50°

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12.

Given: 45°, 90°, find $ x^\circ $

It’s a right triangle with a right angle (90°) and one angle 45°.

So:
$$
x = 180^\circ - 90^\circ - 45^\circ = 45^\circ
$$

But wait — $ x^\circ $ is marked outside at the top, so it's an exterior angle.

The interior angle at the top is 45°, so:
$$
x = 180^\circ - 45^\circ = 135^\circ
$$

Wait — no. The triangle has:
- Right angle: 90°
- One angle: 45°
- So third interior angle: $ 180^\circ - 90^\circ - 45^\circ = 45^\circ $

Now, $ x^\circ $ is the exterior angle at the top (where the 45° interior angle is).

So:
$$
x = 180^\circ - 45^\circ = 135^\circ
$$

✔ Answer: 135°

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | 65° |
| 2 | 63° |
| 3 | 36° |
| 4 | 113° |
| 5 | $ x = 70^\circ $, $ y = 60^\circ $ |
| 6 | 39° |
| 7 | 138° |
| 8 | $ x = 102^\circ $, $ q = 78^\circ $ |
| 9 | $ x = 150^\circ $, $ y = 30^\circ $ |
| 10 | 45° |
| 11 | 50° |
| 12 | 135° |

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